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On a converse to Banach's Fixed Point Theorem

Author: Márton Elekes
Journal: Proc. Amer. Math. Soc. 137 (2009), 3139-3146
MSC (2000): Primary 54H25, 47H10, 55M20, 03E15, 54H05; Secondary 26A16
Published electronically: April 30, 2009
MathSciNet review: 2506473
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Abstract: We say that a metric space $ (X,d)$ possesses the Banach Fixed Point Property (BFPP) if every contraction $ f:X\to X$ has a fixed point. The Banach Fixed Point Theorem states that every complete metric space has the BFPP. However, E. Behrends pointed out in 2006 that the converse implication does not hold; that is, the BFPP does not imply completeness; in particular, there is a nonclosed subset of $ \mathbb{R}^2$ possessing the BFPP. He also asked if there is even an open example in $ \mathbb{R}^n$, and whether there is a `nice' example in $ \mathbb{R}$. In this note we answer the first question in the negative, the second one in the affirmative, and determine the simplest such examples in the sense of descriptive set theoretic complexity.

Specifically, first we prove that if $ X\subset\mathbb{R}^n$ is open or $ X\subset\mathbb{R}$ is simultaneously $ F_\sigma$ and $ G_\delta$ and $ X$ has the BFPP, then $ X$ is closed. Then we show that these results are optimal, as we give an $ F_\sigma $ and also a $ G_\delta$ nonclosed example in $ \mathbb{R}$ with the BFPP.

We also show that a nonmeasurable set can have the BFPP. Our non-$ G_\delta$ examples provide metric spaces with the BFPP that cannot be remetrized by any compatible complete metric. All examples are in addition bounded.

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Márton Elekes
Affiliation: Rényi Alfréd Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary – and – Eötvös Loránd University, Budapest, Hungary

Received by editor(s): February 2, 2007
Received by editor(s) in revised form: March 26, 2007
Published electronically: April 30, 2009
Additional Notes: The author was partially supported by Hungarian Scientific Foundation grants no. 43620, 49786, 61600, 72655, and the János Bolyai Fellowship.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.